Generalized radix representations and dynamical systems III

Abstract

For $\mathbf{r}=(r_{1},\ldots,r_{d}) \in\mathbb{R}^{d}$ the map $\tau_{\mathbf{r}}\colon \mathbb{Z}^{d} \to \mathbb{Z}^{d}$ given by \[ \tau_{\mathbf{r}}(a_{1},\ldots,a_{d}) =(a_{2},\ldots,a_{d},-\lfloor r_{1}a_{1}+ \cdots + r_{d}a_{d} \rfloor) \] is called a shift radix system if for each $\mathbf{a} \in \mathbb{Z}^{d}$ there exists an integer $k>0$ with $\tau_{\mathbf{r}}^{k}(\mathbf{a})=0$. As shown in the first two parts of this series of papers shift radix systems are intimately related to certain well-known notions of number systems like $\beta$-expansions and canonical number systems. In the present paper further structural relationships between shift radix systems and canonical number systems are investigated. Among other results we show that canonical number systems related to polynomials \[ \sum_{i=0}^{d} p_{i} X^{i} \in \mathbb{Z}[X] \] of degree $d$ with a large but fixed constant term $p_{0}$ approximate the set of ($d-1$)-dimensional shift radix systems. The proofs make extensive use of the following tools: Firstly, vectors $\mathbf{r} \in\mathbb{R}^{d}$ which define shift radix systems are strongly connected to monic real polynomials all of whose roots lie inside the unit circle. Secondly, geometric considerations which were established in Part I of this series of papers are exploited. The main results establish two conjectures mentioned in Part II of this series of papers.